MODEL ANSWERS TO HWK #2
(18.022 FALL 2010)
(1) (1.3.27)
(a) First let us x a labeling for the points: W1 , W2 and W3 are the centers of what we will
call the circles 1, 2 and 3, respectively, O is the point where all three circles intersect,
and A, B and

Practice Exam 2 Solutions
Problem 1. Let f : R2 R be a scalar eld. For each of the following questions,
answer yes or no. If the answer is yes, cite a theorem or give a brief sketch
of a proof. If the answer is no, provide a counterexample.
1. Suppose f (

Practice Exam 1 Solutions
Problem 1. Let A be an m n matrix and r be the rank of A.
1. Describe the dimension of the solution space of the equation Ax = 0 in terms
of m, n, r.
2. Suppose there exists c such that Ax = c does not have a solution. What can
y

THIRD MIDTERM
MATH 18.022, MIT, AUTUMN 10
You have 50 minutes. This; host; is Closed book, closed notes, 110 calculators.
Name: If [ODEL NS BEES
Signature:
Recitation Time:
There are 5 problems, and the total number of points is 100. Show
all y

SECOND MIDTERM
MATH 18.022, MIT, AUTUMN 10
You have 50 minutes. This test is closed book, closed notes, no calculators.
Name:_\)jl_5§-:L.&\Lr2£S
Signature:
Recitation Time:
There are 5 problems, and the total number of points is 100. Show

Notes - double integrals.
on (Read 11.1-11.5 of Apostol.)
Just as for the case of a single integral, we have the
following condition for the existence of a double integral:
Theorem 1 (Riemann condition). Q = [arb] x [c,d] given any s
E
.
Then ther

Derivatives - vector functions.
of Recall that if x is a point of a scalar function of x, R "
and if f
f ( 5 ) is (if it
then the derivative of
exists) is the vector
For some purposes, it will be convenient to denote the derivative
of
f
by a row mat

The -
inverse of a matrix
We now consider the pro5lern of the existence of multiplicatiave
inverses for matrices. A t this point, we must take the non-commutativity of matrix.multiplicationinto account.Fc;ritis perfectly possible, given a matrix A, that

Matrices We have already defined what we mean by a matrix. we introduce algebraic operations into the set of matrices. Definition. If A
k by
In this section,
and B are two matrices of the same size, say n matrix obtained by adding
n,
we define A
t
B to be

StokesI Theorem
Our text states and proves Stokes' Theorem in 12.11, but ituses the scalar form for writing both the line integral and the surface integral
involved. In the applications, it is the vector form of the theorem that is
most likely to be q